Fixed points of endomorphisms of trace monoids

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. If the dependence alphabet is a transitive forest, it is proved that the set of regular fixed points of the (Scott) continuous extension of an endomorphism to real traces is Ω-rational for every endomorphism if and only if the monoid is a free product of free commutative monoids.     

Sofic systems

A symbolic flow is called a sofic system if it is a homomorphic image (factor) of a subshift of finite type. We show that every sofic system can be realized as a finite-to-one factor of a subshift of finite type with the same entropy. From this it follows that sofic systems share many properties with subshifts of finite type. We concentrate especially on the properties of TPPD (transitive with periodic points dense) sofic systems.     

Transitive points via Furstenberg family

Let (X,T) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z₊ with hereditary upward property). A point x∈X is called an F-transitive one if {n∈Z₊:Tnx∈U}∈F for every non-empty open subset U of X; the system (X,T) is called F-point transitive if there exists some F-transitive point. In this paper, we aim to classify transitive systems by F-point transitivity. Among other things, it is shown that (X,T) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive).It is shown that every weakly mixing system is Fip-point transitive, while we construct an Fip-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ^?(Fwt)-transitive if and only if it is weakly disjoint from every P-system.     

On the approximation of time one maps of Anosov flows by Axiom A diffeomorphisms

We prove that if 𝒻₁ is the time one map of a transitive and codimension one Anosov flow φ and it is C ¹-approximated by Axiom A diffeomorphisms satisfying a property called P, then the flow is topologically conjugated to the suspension of a codimension one Anosov diffeomorphism. A diffeomorphism 𝒻 satisfies property P if for every periodic point in M the number of periodic points in a fundamental domain of its central manifold is constant. Received: 15 March 2001     

Torsion-free weakly transitive E-Engel Abelian groups

It is proved that if all the endomorphisms of a reduced torsion-free weakly transitive Abelian group are bounded right-nilpotent, then its ring of endomorphisms is commutative. The ring of endomorphisms of a torsion-free Abelian group with periodic group of automorphisms and Engel ring of endomorphisms is also commutative.     

半群作用Li-Yorke对的存在性

设X为紧度量空间,T为半群,本文研究了动力系统（X,T）上Li-Yorke对的存在性问题,证明了当（X,T）拓扑可迁且包含周期点时,在（X,T）上存在无限scrambled集.另外,列举了一些不包含Li-Yorke对的动力系统.     

Polygonal invariant curves for a planar piecewise isometry

We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.

     

拟弱几乎周期点的等价定义与系统的混沌性

1992年, 周作领引进了弱几乎周期点这一概念. 1995年, 周和何伟弘又引进了拟弱几乎周期点这个概念, 并利用它们深刻地刻画了一个动力系统的本质所在. 为了更好地看出这两者的区别,首先从回复时间集的角度给出拟弱几乎周期点的等价定义,然后研究了一个存在真的拟弱几乎周期点的系统的混沌情况,得到了这样的系统是Takens-Ruelle混沌的.     

Sensitivity and chaos of semigroup actions

We prove that if an action of a C-semigroup S on a Polish space is syndetic transitive, then the system is either minimal and equicontinuous, or sensitive. Additionally, we show that if an action of an abelian monoid S on a Polish space has a transitive point x and a periodic orbit O such that [`(Hx)]\overline{Hx} is perfect where H={s∈S:s| _O is an identity map}, then the system is chaotic.     

On transitive one-factorizations of arc-transitive graphs

It is shown that transitive 1-factorizations of arc-transitive graphs exist if and only if certain factorizations of their automorphism groups exist. This relation provides a method for constructing and characterizing transitive 1-factorizations for certain classes of arc-transitive graphs. Then a characterization is given of 2-arc-transitive graphs which have transitive 1-factorizations. In this characterization, some 2-arc transitive graphs and their transitive 1-factorizations are constructed.     

Topologically transitive subsets of piecewise monotonic maps,which contain no periodic points

If one splits the nonwandering set of a piecewise monotonic map into maximal subsets, which are topologically transitive, one gets two kinds of subsets. The first kind of these subsets has periodic orbits dense, the second kind contains no periodic orbits. In this paper it is shown, that there are only finitely many subsets of the second kind, each of which is minimal and has only finitely many ergodic invariant Borel probability measures.     

Shadowing with chain transitivity

A transitive dynamical system is either sensitive or has a dense set of equicontinuity points [E. Akin, J. Auslander, K. Berg, When is a transitive map chaotic, in: Convergence in Ergodic Theory and Probability, Walter de Gruyter & Co., 1996, pp. 25-40]. We show that if a chain transitive system has shadowing property then it is either sensitive or all points are equicontinuous.     

Transitivity and Dense Periodicity for Graph Maps

In 1984, Blokh proved [A. M. Blockh, On transitive mappings of one-dimensional branched manifolds, Differential-Difference Equations and Problems of Mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 131, pp. 3–9, 1984] that any topologically transitive continuous map from a graph into itself which has periodic points has a dense set of periodic points and has positive topological entropy (in this proof a crucial role is played by the specification property, which implies these two statements). Also, he characterized the topologically transitive continuous graph maps without periodic points. Unfortunately, this clever paper is only available in Russian (except for a translation to English of the statements of the theorems without proofs—see [A. M. Blockh, The connection between entropy and transitivity for one-dimensional mappings, Uspekhi Mat. Nauk, 42(5(257)) (1987), pp. 209–210]).     

On Finite 2‐Path‐Transitive Graphs

The class of graphs that are 2‐path‐transitive but not 2‐arc‐transitive is investigated. The amalgams for such graphs are determined, and structural information regarding the full automorphism groups is given. It is then proved that a graph is 2‐path‐transitive but not 2‐arc‐transitive if and only if its line graph is half‐arc‐transitive, thus providing a method for constructing new families of half‐arc‐transitive graphs. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 225–237, 2013     

A structure of doubly stochastic Markov chains

Kingman and Williams [6] showed that a pattern of positive elements can occur in a transition matrix of a finite state, nonhomogeneous Markov chain if and only if it may be expressed as a finite product of reflexive and transitive patterns. In this paper we solve a similar problem for doubly stochastic chains. We prove that a pattern of positive elements can occur in a transition matrix of a doubly stochastic Markov chain if and only if it may be expressed as a finite product of reflexive, transitive, and symmetric patterns. We provide an algorithm for determining whether a given pattern may be expressed as a finite product of reflexive, transitive, and symmetric patterns. This result has implications for the embedding problem for doubly stochastic Markov chains. We also give the application of the obtained characterization to the chain majorization.     

On geometrically transitive Hopf algebroids

This paper contributes to the characterization of a certain class of commutative Hopf algebroids. It is shown that a commutative flat Hopf algebroid with a non zero base ring and a nonempty character groupoid is geometrically transitive if and only if any base change morphism is a weak equivalence (in particular, if any extension of the base ring is Landweber exact), if and only if any trivial bundle is a principal bi-bundle, and if and only if any two objects are fpqc locally isomorphic. As a consequence, any two isotropy Hopf algebras of a geometrically transitive Hopf algebroid (as above) are weakly equivalent. Furthermore, the character groupoid is transitive and any two isotropy Hopf algebras are conjugated. Several other characterizations of these Hopf algebroids in relation to transitive groupoids are also given.     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.     

Linear systems with removable base loci

Abstract

We develop several local approaches for the three classes of finite groups: T-groups (normality is a transitive relation) and PT-groups (permutability is a transitive relation) and PST-groups (S-permutability is a transitive relation). Here a subgroup of a finite group G is S-permutable if it permutes with all the Sylow subgroup of G.     

Local 2-Geodesic Transitivity of Graphs

An s-geodesic in a graph Γ is a path connecting two vertices at distance s. Being locally transitive on s-geodesics is not a monotone property: if an automorphism group G of a graph Γ is locally transitive on s-geodesics, it does not follow that G is locally transitive on shorter geodesics. In this paper, we characterise all graphs that are locally transitive on 2-geodesics, but not locally transitive on 1-geodesics.     

Innately transitive subgroups of wreath products in product action

A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.